Built on Facts

Sunday Function

Here’s this week’s Sunday Function. In the universe of functions it’s an utterly typical suburban middle class citizen, with a pleasant but quite ordinary job in a downtown cubicle farm for Physics Incorporated.

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His name is
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In a little more detail, you’ll notice that this function is the product of two elementary functions. There’s x squared, and there’s e raised to the negative x power. As x increases, obviously x squared increases quite quickly. By the time you get to x = 500, x squared is equal to 250,000. But on the other hand, e^-x is shrinking as x increases. It happens to be shrinking even faster than x squared is growing. By the time x = 500, e^-x is equal to approximately a mere 0.000…00712, where there’s a total of 217 zeros to the right of the decimal point. The shrinking exponential “wins” and as x gets large, f(x) gets small.

This is a reflection of a more general mathematical truth. If we has x cubed instead of x squared, the function would still go to zero as x increased. The negative exponential still wins. In fact, any constant power of x is going to be “beaten” by the exponential, which grows faster than any power of x.

These kinds of relationships are very important in physics. It may be very hard to pry apart the specifics of some complicated function, but if you recognize which parts dominate the function for large x, you may be making some very important headway. The general name for this type of examination goes by the rather suggestive moniker of Big O notation.

In physics we don’t usually come across functions which grow faster than the negative exponential shrinks, but they do exist. The factorial function is one of the most commonly encountered.

Pure mathematics has functions which just grow incomprehensibly big at an astonishing rate. Here’s an excellent and very readable essay on the subject of large numbers by Scott Aaronson.

It’s a nice way to realize that even the simple tools of math can quickly be extended into some truly arcane territory.

Comments

  1. #1 dWj
    December 28, 2008

    What does it do? The first “use” that jumps out at me is that this is (if you normalize it) the Poisson distribution with mean 2. It’s not quite the blackbody energy distribution if I remember correctly, but it wouldn’t surprise me if there were some thermal system to which this were relevant (that didn’t look contrived).

  2. #2 Comrade PhysioProf
    December 28, 2008

    Coolio!! Does the x!e[-x] function have some physical significance?

  3. #3 Matt Springer
    December 28, 2008

    This particular type of function (a polynomial times e^-x) is also more or less the radial wavefunction for the hydrogen atom.

    Physio, I can’t think of one offhand but it wouldn’t surprise me if it did.

  4. #4 joemac53
    December 28, 2008

    When you run into one of my high-school students they will tell you that e^(-x) “beats the snot out of” any power of x. (they might even use coarser language)

  5. #5 mike
    December 28, 2008

    Wow, I just read that paper on big numbers, it was quite inspiring.

  6. #6 dean
    December 28, 2008

    “Poisson distribution with mean 2.”
    Not directly – the function as given is continuous, but the Poisson is discrete. You have to normalize it and restrict it to the integers to get the Poisson distribution.

  7. #7 Tim
    December 29, 2008

    Physioprof, I don’t think x! * e^(-x) would have any significance – x! is basically x^x * e^(-x). For large x, the function would blow up very quickly, even with the extra e^(-x).

  8. #8 Blaise Pascal
    December 29, 2008

    The Planck distribution is similar, with the added complication that it’s a two-variable function (in ν and T). It’s I(ν,T) = aν3/(ebν/T-1), where a and be are constants.

    Holding T constant, and realizing that 1/(ex-1) is asymptotically close to e-x, then the Planck distribution is approximately aν3e-bν.

    (hey! my nu’s got replaced by v’s… what gives?)

  9. #9 agm
    January 6, 2009

    Gotta love L’Hopital’s rule, which is how ya tell which factor in a function dominates in a limit like this.